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Properties

Label 9.9.b
Level $9$
Weight $9$
Character orbit 9.b
Rep. character $\chi_{9}(8,\cdot)$
Character field $\Q$
Dimension $2$
Newform subspaces $1$
Sturm bound $9$
Trace bound $0$

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Defining parameters

Level: \( N \) \(=\) \( 9 = 3^{2} \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 9.b (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 3 \)
Character field: \(\Q\)
Newform subspaces: \( 1 \)
Sturm bound: \(9\)
Trace bound: \(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{9}(9, [\chi])\).

Total New Old
Modular forms 10 2 8
Cusp forms 6 2 4
Eisenstein series 4 0 4

Trace form

\( 2 q + 476 q^{4} + 3304 q^{7} + O(q^{10}) \) \( 2 q + 476 q^{4} + 3304 q^{7} - 8388 q^{10} - 52544 q^{13} + 104072 q^{16} + 93280 q^{19} + 181296 q^{22} - 1173154 q^{25} + 786352 q^{28} - 392888 q^{31} - 128844 q^{34} + 5638828 q^{37} - 4143672 q^{40} - 4426928 q^{43} + 2783952 q^{46} - 6071394 q^{49} - 12505472 q^{52} + 42241968 q^{55} + 5218308 q^{58} - 34810604 q^{61} + 20216432 q^{64} - 28645328 q^{67} - 13856976 q^{70} - 17813984 q^{73} + 22200640 q^{76} + 65517688 q^{79} + 5994252 q^{82} - 30020652 q^{85} + 89560224 q^{88} - 86802688 q^{91} - 13856112 q^{94} - 48903488 q^{97} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{9}^{\mathrm{new}}(9, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
9.9.b.a 9.b 3.b $2$ $3.666$ \(\Q(\sqrt{-2}) \) None \(0\) \(0\) \(0\) \(3304\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta q^{2}+238q^{4}+233\beta q^{5}+1652q^{7}+\cdots\)

Decomposition of \(S_{9}^{\mathrm{old}}(9, [\chi])\) into lower level spaces

\( S_{9}^{\mathrm{old}}(9, [\chi]) \cong \) \(S_{9}^{\mathrm{new}}(3, [\chi])\)\(^{\oplus 2}\)